Maine
As concluded from the initial summary statistics, we start off with Maine as a test site. The location we chose for Maine is 3 kilometers offshore with reliable wave and wind currents and thus, a good starting point to test our ability to forecast tidal power.
We proceeded with looking at the time series plots for both daily and monthly mean wave power. We also looked at the ACF and PACF plots of both series, so that we can make an informed decision on which time series (daily or monthly) to pursue.
From Figure 1, we see that while the daily series provides a more granular view of the fluctuations in mean wave power, it is subject to extreme outliers that can affect our forecasting ability. Additionally, when compared to the monthly series, there is a lot more noise in the graph, making it harder to discern trends.
The monthly series on the other hand shows a clearer seasonal pattern. Further, by averaging out to a monthly series, the impact of outliers is diminished.
Figure 1. Plots for mean wave power in Maine (Daily and Monthly)
From Figure 2, we see that both ACF and PACF plots for the daily series have a sharp drop-off after lag 1, suggesting that past daily mean wave power values do not carry much signal for future values. For the monthly mean wave power, the ACF and PACF plots shows strong seasonality with the wave pattern. The autocorrelation persists over time, which will be ideal for ARIMA/SARIMA and other seasonal models.
Preparing the data
Thus, we will proceed with forecasting for the monthly series. We start by splitting our monthly mean wave power datasets using the 80-20 rule: 80% for training, 20% for testing. Our training dataset for Maine contains mean wave power from January 1979 to December 2003 and our testing dataset for Maine contains mean wave power from January 2004 to December 2010.
We used the decompose() function on our training dataset and performed an additive deomposition. From Figure 3, we can confirm the strong seasonality seen in mean wave power in Maine. We also see that mean wave power peaked from 1995 to around 2000 in Maine. This could be due to climate and oceanographic factors (wind speed, underwater currents) that warrant a further look. Since tidal energy is subject to a whole host of climate and oceanographic factors, our remainder series shows some significant spikes at irregular intervals which could be caused by specific weather events. Further research into Maine’s short-term and long-term weather patterns is recommended for future forecasting projects.
Lastly, we ran a ADF test on our training dataset on Maine, which produced a p-value of 0.01. Thus, we reject the null hypothesis that the Maine monthly mean wave power series has a unit root and is non-stationary.
Start of forecasting models
We proceed by setting the forecasting length to be from January 2004 to December 2010 (testing dataset). We will evaluate our forecasts against values found in our testing datasets.
For forecasting, we trained and tested the following models: - Seasonal Naïve: Used as a preliminary baseline due to strong seasonal patterns in monthly wave power
ARIMA + Fourier: Applied using auto.arima() function with a fourier term
ETS + STL: Implements seasonal trend decomposition (Figure 3) and performs exponential smoothing using stlf() function
SARIMA: Applied a seasonal ARIMA model (due to strong seasonality present) using the auto.arima() function
ES: Exponential smoothing to use recent, past values of monthly wave power to forecast future ones
StructTS: Uses a state-space framework to model unobserved components of monthly tidal series. Applied using a Basic Structural Model (BSM).
TBATS: Used to model complex seasonal patterns present in tidal power
Neural Network + XREG: Single, hidden-layer neural network (with our training dataset as an external regressor).
Table 1 displays our accuracy metrics for our models.
Judging from the RMSE and MAPE, our 3 best models are STL + ETS, TBATS and ARIMA + Fourier. This is in line with our hypothesis that models that capture the complex seasonality of tidal power would perform best.
| ME | RMSE | MAE | MPE | MAPE | ACF1 | Theil’s U | |
|---|---|---|---|---|---|---|---|
| SNAIVE | -822.52074 | 4290.692 | 3202.526 | -19.62825 | 42.60721 | 0.06549 | 0.89498 |
| SARIMA | -217.59468 | 4026.117 | 3221.739 | -29.70686 | 50.01054 | 0.32049 | 0.83462 |
| STL+ETS | -192.97893 | 3730.735 | 2709.232 | -17.21843 | 36.39588 | 0.18831 | 0.70042 |
| ARIMA+Fourier | 736.38317 | 3690.025 | 2601.569 | -5.30294 | 31.99013 | 0.20083 | 0.72561 |
| ES | 4686.01400 | 5864.526 | 4762.718 | 78.82107 | 80.54519 | -0.00122 | 1.84991 |
| StructTS | 4686.01400 | 5864.526 | 4762.718 | 78.82107 | 80.54519 | -0.00122 | 1.84991 |
| TBAT | 569.08601 | 3728.780 | 2690.142 | -8.37019 | 33.99924 | 0.17281 | 0.70592 |
| NN | 79.13025 | 4234.965 | 3058.354 | -11.06327 | 38.95378 | 0.08974 | 0.78838 |
Figures 4 and 5 look at how our models forecast monthly mean wave power for Maine for our top 3 models, and compares it with actuals. We see that all 3 models capture the seasonality of tidal power well. However, they also underestimate its highest peaks on average. This could likely be due to not accounting for the many exogenous variables that can cause spikes in tidal power (powerful rip currents, strong weather storms, sudden changes in wind velocity). Our 3 models are dependable in providing a minimum power generation forecast, which is helpful for energy planners and policymakers who want to understand the cost-benefit analysis of harnessing tidal energy. Additionally, the underestimation of maximum capacity can be beneficial when building power stations as the forecasts can be used to determine safe lower bounds on maximum capacity.
Figure 4. Forecast comparisons of our top 3 models
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Figure 5. Comparing Forecasted Values with Actuals - Maine
Given the performance of our 3 chosen models, we proceed with fitting them on Alaska and Florida.